Calculate $\int_{S}x_1^3dvol_{n-1}$ where $S=\{x\in\mathbb{S}^{n-1}:x_1>0\}$
$\int_{S}x_1^3dvol_{n-1}=\int_{S}<F,N>=[F=(x_1^2,0\ldots,0),N=x]=\int_{B=\{x\in\mathbb{R}^n:x\in B(0,1),x_1>0\}}2x_1dx$
I'm not sure how to continue from here hints will be welcome
By invoking the parameterization $$\vec{r}(x_1,\ldots,x_{n-1})=\Big(\sqrt{1-x_1^2-\ldots-x_{n-1}^2},x_1,\ldots,x_{n-1}\Big)$$ we see that $$\begin{eqnarray*}\int_{S}x_1^3\mathrm{d}V_{n-1}&=&\int_{\{x_1^2+\dots+x_{n-1}^2<1\}}(1-x_1^2-\dots-x_{n-1}^2)\mathrm{d}V_{n-1} \\ &=& \int_{\{x_1^2+\dots x_{n-1}^2<1\}}g\Big(\sqrt{x_1^2+\dots + x_{n-1}^2}\Big)\mathrm{d}V_{n-1} \\ &=& \int_0^1g(r)A_{n-1}(r)\mathrm{d}r \end{eqnarray*}$$ where $g(x)=1-x^2$ and $A_n(x)=\frac{n \pi^{n/2}r^{n-1}}{(n/2)!}$. Here, $A_n(x)$ is the surface area of an $n$ sphere of radius $x$. We get with basic integration that $$\int_{S}x_1^3\mathrm{d}V_{n-1}=\frac{(n-1)\pi^{\frac{n-1}{2}}}{\big(\frac{n-1}{2}\big)!}\Bigg[\frac{1}{n-1}-\frac{1}{n+1}\Bigg]$$